Thermodynamics of the Dead-Zone Inner Edge in Protoplanetary Disks⋆

A&A 564, A22 (2014) Astronomy DOI: 10.1051/0004-6361/201321911 & c ESO 2014 Astrophysics

Thermodynamics of the dead-zone inner edge in protoplanetary disks

Julien Faure1, Sébastien Fromang1, and Henrik Latter2

1 Laboratoire AIM, CEA/DSM – CNRS – Université Paris 7, Irfu/Service d’Astrophysique, CEA-Saclay, 91191 Gif-sur-Yvette, France e-mail: [email protected] 2 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK Received 17 May 2013 / Accepted 20 February 2014 ABSTRACT

Context. In protoplanetary disks, the inner boundary between the turbulent and laminar regions could be a promising site for planet formation, thanks to the trapping of solids at the boundary itself or in vortices generated by the Rossby wave instability. At the interface, the disk thermodynamics and the turbulent dynamics are entwined because of the importance of turbulent dissipation and thermal ionization. Numerical models of the boundary, however, have neglected the thermodynamics, and thus miss a part of the physics. Aims. The aim of this paper is to numerically investigate the interplay between thermodynamics and dynamics in the inner regions of protoplanetary disks by properly accounting for turbulent heating and the dependence of the resistivity on the local temperature. Methods. Using the Godunov code RAMSES, we performed a series of 3D global numerical simulations of protoplanetary disks in the cylindrical limit, including turbulent heating and a simple prescription for radiative cooling. Results. We find that waves excited by the turbulence significantly heat the dead zone, and we subsequently provide a simple theoretical framework for estimating the wave heating and consequent temperature profile. In addition, our simulations reveal that the dead-zone inner edge can propagate outward into the dead zone, before stalling at a critical radius that can be estimated from a mean- field model. The engine driving the propagation is in fact density wave heating close to the interface. A pressure maximum appears at the interface in all simulations, and we note the emergence of the Rossby wave instability in simulations with extended azimuth. Conclusions. Our simulations illustrate the complex interplay between thermodynamics and turbulent dynamics in the inner regions of protoplanetary disks. They also reveal how important activity at the dead-zone interface can be for the dead-zone thermodynamic structure. Key words. magnetohydrodynamics (MHD) – turbulence – protoplanetary disks

1. Introduction other hand, the interface will influence the radial profiles of the dead zone’s thermodynamic variables – temperature and en- Current models of protoplanetary (PP) disks are predicated on tropy, most of all. Not only will it affect the global disk struc- the idea that significant regions of the disk are too poorly ion- ture and key disk features (such as the ice line), but the interface ized to sustain magneto-rotational-instability (MRI) turbulence. will also control the preconditions for dead-zone instabilities that These PP disks are thought to comprise a turbulent body of feed on the disk’s small adverse entropy gradient, such as the plasma (the “active zone”) enveloping a region of cold quies- subcritical baroclinic instability and double-diffusive instability cent gas, in which accretion is actually absent (the “dead zone”) (Lesur & Papaloizou 2010; Latter et al. 2010). (Gammie 1996; Armitage 2011). These models posit a critical Most studies of the interface have been limited to isother- ∼ inner radius ( 1 au) within which the disk is fully turbulent and mality. This is a problematic assumption because of the perva- beyond which the disk exhibits turbulence only in its surface lay- sive interpenetration of dynamics and thermodynamics in this − ers, for a range of radii (1 10 au) (but see Bai & Stone 2013, for region, especially at the midplane. Temperature depends on the a complication of this picture). turbulence via the dissipation of its kinetic and magnetic fluc- The inner boundary between the MRI-active and dead re- tuations, but the MRI turbulence, in turn, depends on the tem- gions is crucial for several key processes. Because there is a perature through the ionization fraction, which is determined by mismatch in accretion across the boundary, a pressure maximum thermal ionization (Pneuman & Mitchell 1965; Umebayashi & will naturally form at this location, which (a) may halt the in- Nakano 1988). Because of this feedback loop, the temperature ward spiral of centimetre to metre-sized planetesimals (Kretke is not an additional piece of physics that we add to simply com- et al. 2009); and (b) may excite a large-scale vortex instabil- plete the picture; it is instead at the heart of how the interface ity (“Rossby wave instability”) (Lovelace et al. 1999)thatmay and its surrounding region work. One immediate consequence promote dust accumulation, hence planet formation (Barge & of this feedback is that much of the midplane gas inward of 1 au Sommeria 1995; Lyra et al. 2009; Meheut et al. 2012). On the is bistable: if the gas at a certain radius begins as cold and poorly ionized, it will remain so; conversely, if it begins hot and turbu- Appendices are available in electronic form at lent, it can sustain this state via its own waste heat (Latter & http://www.aanda.org Balbus 2012). Thus two stable states are available at any given Article published by EDP Sciences A22, page 1 of 15 A&A 564, A22 (2014) radius in the bistable region. This complicates the question of prescriptions, main parameters, and initial and boundary condi- where the actual location of the dead zone boundary lies. It also tions. In order to keep the discussion as general as possible, all raises the possibility that the boundary is not static, and may not variables and equations in this section are dimensionless. even be well defined. Similar models have also been explored in the context of FU Ori outbursts Zhu et al. (2010, 2009a,b). In this paper we simulate these dynamics directly with a set 2.1. Equations of numerical experiments of MRI turbulence in PP disks. We Since we are interested in the interplay between dynamical and ∼ − concentrate on the inner radii of these disks ( 0.1 1au)sothat thermodynamical effects at the dead zone inner edge, we solve our simulation domain straddles both the bistable region and the the MHD equations, with (molecular) Ohmic diffusion, along- inner dead-zone boundary. Our aim is to understand the inter- side an energy equation. We neglect the kinematic viscosity, that actions between real MRI turbulence and thermodynamics, and is much smaller than Ohmic diffusion, because of its minor role thus test and extend previous exploratory work that modelled in the MRI dynamics. However dissipation of kinetic energy is ff the former via a crude di usive process (Latter & Balbus 2012). fully captured by the numerical grid. We adopt a cylindrical co- Our 3D magnetohydrodynamics (MHD) simulations employ the ordinate system (R,φ,Z) centred on the central star: Godunov code RAMSES (Teyssier 2002; Fromang et al. 2006), in which the thermal energy equation has been accounted for ∂ρ ff + ∇·(ρu) = 0(1) and magnetic di usivity appears as a function of temperature ∂t (according to an approximation of Saha’s law). We focus on the ∂ρu + ∇· ρuu − + ∇ = −ρ∇Φ optically thick disk midplane, and thus omit non-thermal sources ∂ ( BB) P (2) of ionization. This also permits us to model the disk in the cylin- t ∂E drical approximation. + ∇· (E + P)u − B(B·u) + Fη = −ρu ·∇Φ −L (3) This initial numerical study is the first in a series, and thus ∂t lays out our numerical tools, tests, and code-checks. We also ∂B − ∇×(u × B) = −∇×(η∇×B)(4) verify previous published results for fully turbulent disks and ∂t disks with static dead zones (Dzyurkevich et al. 2010; Lyra & Mac Low 2012). We explore the global radial profiles of ther- where ρ is the density, u is the velocity, B is the magnetic field, modynamic variables in such disks, and the nature of the turbu- and P is the pressure. Φ is the gravitational potential. In the lent temperature fluctuations, including a first estimate for the cylindrical approximation, it is given by Φ=−GM/R where magnitude of the turbulent thermal flux. We find that the dead G is the gravitational constant and M is the stellar mass. In zone can be effectively heated by density waves generated at the RAMSES, E is the total energy, i.e. the sum of kinetic, magnetic dead zone boundary, and we estimate the resulting wave damp- and internal energy eth (but it does not include the gravitational ing and heating. Our simulations indicate that the dead zone may energy). Since we use a perfect gas equation of state to close be significantly hotter than most global structure models indicate the former set of equations, the latter is related to the pressure = / γ − γ = . because of this effect. Another interesting result concerns the dy- through the relation eth P ( 1) in which 1 4. The mag- namics of the dead zone interface itself. We verify that the inter- netic diffusivity is denoted by η. As we only consider thermal face is not static and can migrate from smaller to larger radii. ionization, η will depend on temperature T; its functional form All such simulated MRI “fronts” ultimately stall at a fixed ra- we discuss in the following section. Associated with that resistiv- dius set by the thermodynamic and radiative profiles of the disk, ity is a resistive flux Fη that appears in the divergence of Eq. (3). in agreement with Latter & Balbus (2012). These fronts move Its expression is given by Eq. (23) of Balbus & Hawley (1998). more quickly than predicted because they propagate not via the The L symbol denotes radiative losses and it will be described in slower MRI turbulent motions but by the faster density waves. Sect. 2.3. Dissipative and radiative cooling terms are computed Finally, we discuss instability near the dead zone boundary. using an explicit scheme. This method is valid if the radiative The structure of the paper is as follows. In Sect. 2 we de- cooling time scale is much longer than the typical dynamical scribe the setup we used for the MHD simulations and give our time(seeSect.2.4). Finally, we have added a source term in the prescriptions for the radiative cooling and thermal ionization. continuity equation that maintains the initial radial density pro- ρ We test our implementation of thermodynamical processes in file 0 (Nelson & Gressel 2010; Baruteau et al. 2011). It is such models of fully turbulent disks in Sect. 3. There we also quan- that tify the turbulent temperature fluctuations and the turbulent heat ∂ρ ρ − ρ flux. Section 4 presents results of resistive MHD simulations = − 0 · (5) with dead zones. Here we look at the cases of a static and dy- ∂t τρ namic dead zone separately. Rossby wave and other instabilities are briefly discussed in Sect. 5; we subsequently summarize our The restoring time scale τρ is set to 10 local orbits and prevents results and draw conclusions in Sect. 6. the long term depletion of mass caused by the turbulent transport through the inner radius. 2. Setup 2.2. Magnetic diffusion We present in this paper a set of numerical simulations performed using a version of the code RAMSES (Teyssier 2002; In this paper, we conduct both ideal (η = 0) and non-ideal (η Fromang et al. 2006) which solves the MHD equations on a 0) simulations. In the latter case, we treat the disk as partially 3D cylindrical and uniform grid. Since we are interested in ionized. In such a situation, the magnetic diffusivity is known the inner-edge behaviour at the mid-plane, we discount verti- to be a function of the temperature T and ionization fraction xe cal stratification and work under the cylindrical approximation (Blaes & Balbus 1994): (Armitage 1998; Hawley 2001; Steinacker & Papaloizou 2002). η ∝ −1 1/2. In the rest of this section we present the governing equations, xe T (6) A22, page 2 of 15 J. Faure et al.: Dead zone inner edge dynamics

The ionisation fraction can be evaluated by considering the ion- energy is dissipated on the grid; our use of a total energy equa- ization sources of the gas. In the inner parts of PP disks, the mid- tion ensures that this energy is not lost but instead fully trans- plane electron fraction largely results from thermal ionization ferred to thermal energy. It should be noted that though this is (neglecting radioisotopes that are not sufficient by themselves perfectly adequate on long length and time scales, the detailed to instigate MRI). Non thermal contributions due to ionization flow structure on the dissipation scale will deviate significantly by cosmic (Sano et al. 2000), UVs (Perez-Becker & Chiang from reality. 2011) and X-rays (Igea & Glassgold 1999) are negligible because of the large optical thickness of the disk at these radii. In fact, the ionization is controlled by the low ionization poten- 2.4. Initial conditions and main parameters tial (Ei ∼ 1−5 eV) alkali metals, like sodium and potassium, − Our 3D simulations are undertaken on a uniformly spaced grid the abundance of which is of order 10 7 smaller than hydrogen in cylindrical coordinates (R,φ,Z). The grid ranges over R ∈ (Pneuman & Mitchell 1965; Umebayashi & Nakano 1988). Then [R0, 8 R0], φ ∈ [0,π/4] and Z ∈ [−0.3 R0, 0.3 R0]. Here the key xe can be evaluated using the Saha equation. When incorporated length scale R0 serves as the inner radius of our disk domain and into Eq. (6), one obtains the following relation between the re- could be associated with a value between 0.2−0.4 au in physical sistivity and the temperature: units. Each run has a resolution [320, 80, 80], which has been ffi − 1 shown in these conditions to be su cient for MHD turbulence η ∝ T 4 exp (T /T), (7) to be sustained over long time scales (Baruteau et al. 2011). For with T a constant. Because of the exponential, η varies very completeness, we present a rapid resolution study in Appendix D rapidly with T, and, in the context of MRI activation, can be that further strengthens our results. Throughout this paper, we denote by X0 the value of the thought of as a “switch” around a threshold TMRI. In actual pro- = toplanetary disks, it is well known that that temperature is around quantity X at the inner edge of the domain, i.e. at R R0. Units 103 K(Balbus & Hawley 2000; Balbus 2011). Taking this into are chosen such that: account, and for the sake of simplicity, we use in our simulations GM = R =Ω = ρ = T = 1, the approximation: 0 0 0 0 where Ω stands for the gas angular velocity at radius R. Thus η if T < T η(T) = 0 MRI (8) all times are measured in inner orbits, so a frequency of unity 0otherwise, correspond to a period of one inner orbit. We will refer to the local orbital time at R using the variable τorb. where TMRI is the activation temperature for the onset of MRI. η The initial magnetic field is a purely toroidal field whose pro- Note that by taking a step function form for we must neglect file is built to exhibit no net flux and whose maximum strength its derivative in Eq. (4). corresponds to β = 25: π 2.3. Radiative cooling and turbulent heating = 2P 2 · Bφ β sin − Z (10) As we are working within a cylindrical model of a disk, with- Zmax Zmin out explicit surface layers, we model radiative losses using the following cooling function: The simulations start with a disk initially in approximate radial force balance (we neglect the small radial component of L = ρσ T 4 − T 4 . (9) the Lorentz force in deriving that initial state). Random veloc- min ities are added to help trigger the MRI. Density and temperature This expression combines a crude description of radiative cool- profiles are initialized with radial power laws: −ρσ 4 ing of the disk ( T ) as well as irradiation by the central star p q (+ρσT 4 ), in which σ should be thought of as a measure of R R min ρ = ρ0 ; T = T0 , (11) the disk opacity (see also Sect. 2.4). In the absence of any other R0 R0 heating or cooling sources, T should tend toward Tmin, the equilibrium temperature of a passively irradiated disk. In principle, where p and q are free parameters. Pressure and temperature are related by the ideal gas equation of state: Tmin is a function of position and surface density, but we take it to be uniform for simplicity. In practice, we found it has lit- ρ T tle influence on our results. Turbulence or waves ensure that the = , P P0 ρ (12) simulated temperature is always significantly larger than Tmin. 0 T0 The parameter σ determines the quantity of thermal energy the where P depends on the model and is specified below. We gas is able to hold. For computational simplicity and to ease the 0 choose p = −1.5, which sets the initial radial profile for the den- physical interpretation of our results, we take σ to be uniform. sity. During the simulation, it is expected to evolve on the long In reality, σ should vary significantly across the dust sublima- secular time scale associated with the large-scale accretion flow tion threshold (∼1500 K) (Bell & Lin 1994), but an MRI front (except possibly when smaller scale feature like pressure bump will always be much cooler (∼1000 K). As a consequence, the appear, see Sect. 4). Quite differently, the temperature profile strong variation of σ will not play an important role in the dy- evolves on the shorter thermal time scale (which is itself longer namics. Finally, we omit the radiative diffusion of heat in the than the dynamical time scale τ ). It is set by the relaxation planar direction, again for simplicity though in real disks it is an orb time to thermal equilibrium1. important (but complicated) ingredient. ff Our prescription for magnetic di usion, Eq. (8), takes care of 1 One consequence is that the ratio H/R which measures the relative the dissipation of magnetic energy when T < TMRI.However,if importance of thermal and rotational kinetic energy, is no longer an the disk is sufficiently hot the dissipation of magnetic energy (as input parameter, as in locally isothermal simulations, but rather the out- with the kinetic energy) is not explicitly calculated. In this case come of a simulation.

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As shown by Shakura & Sunyaev (1973), turbulent heating defined. We used Tmin = 0.05 T0, which means that the temper- in an accretion disk is related to the Rφ component of the turbu- ature in the hot turbulent innermost disk radius is 20 times that lent stress tensor TRφ: of a cold passive irradiated disk. As discussed in Sect. 2.3, Tmin is so small that it has little effect on our results. The values of ∂Ω T and η vary from model to model and will be discussed in Q+ = −TRφ ∼ 1.5ΩTRφ. (13) MRI 0 ∂ ln R the appropriate sections. In order to assess how realistic our disk model is, we con- For MHD turbulence TRφ amounts to (Balbus & Papaloizou 1999): vert some of its key variables to physical units. This can be done as follows. We consider a protoplanetary disk orbiting around a 3 −2 TRφ = −BRBφ + ρvRδvφ. (14) half solar mass star, with surface density Σ ∼ 10 gcm and temperature T ∼ 1500 K at 0.1 au. (Given the radial profile The . notation stands for an azimuthal, vertical and time aver- δv we choose for the surface density, this would correspond to a age, φ is the flow’s azimuthal deviation from Keplerian rota- disk mass of about seven times the minimum mass solar nebu- tion. The last scaling in Eq. (13) assumes such deviations are lar within 100 au of the central star.) For this set of parameters, small, i.e. the disk is near Keplerian rotation. Balancing that σ = × −9 4 we can use Eq. (18) to calculate 1 10 in cgs units for heating rate by the cooling function L∼ρσT gives a relation −2 model σcold,usingα ∼ 10 . In the simulations, the vertically for the disk temperature (here we neglect Tmin in the cooling integrated cooling rate is thus given by the cooling rate per unit function): surface: 4 σρT 1.5ΩT φ. (15) 4 6 −2 −1 R Q− =ΣσT = 5.3 × 10 erg cm s . (20) Using the standard α prescription for illustrative purposes, we α = α α This value can be compared to the cooling rate of a typical write TRφ P which defines the Shakura-Sunyaev parame- disk with the same parameters (Chambers 2009): ter, a constant in the classical α disk theory (Shakura & Sunyaev Ω ∝ −1.5 ∝ −0.5 σ 1973). Since R ,Eq.(15) suggests that T R in PP 8 b 4 Q− = T , (21) equilibrium for uniform σ values. We thus used q = −0.5 in our 3 τ initialization of T. σ τ = κ Σ/ We can use a similar reasoning to derive an estimate for the where b is the Stefan-Boltzman constant and 0 2in κ = 2 −1 thermal time-scale. Using the turbulent heating rate derived by which 0 1cm g stands for the opacity. Using these fig- PP = . × 6 −2 −1 Balbus & Papaloizou (1999) and the definition of α, the mean ures, we obtain Q− 1 5 10 erg cm s ,i.e.avaluethat internal energy evolution equation can be approximated by is close (given the level of approximation involved) to that used in the simulations2 (see Eq. (20)). We caution that this accept- ∂e P able agreement should not be mistaken as a proof that we are th ∼ = 1.5ΩαP−L, (16) ∂t (γ − 1)τheat correctly modelling all aspects of the disk’s radiative physics. The cooling function we use is too simple to give anything more which yields immediately than an idealized thermodynamical model. It only demonstrates τ−1 ∼ 1.5Ω(γ − 1)α. (17) the consistency between the thermal time scale we introduce in heat the simulations and the expected cooling time scale in proto- For α = 0.01, the heating time scale is thus about 25 local orbits. planetary disks. The cooling time-scale τcool is more difficult to estimate, but near equilibrium should be comparable to τ . heat 2.5. Buffers and boundaries In order to obtain a constraint on the parameter σ we use 2 = γ /ρ ≈ 2Ω2 the relations cs P H ,wherecs is the sound speed Boundary conditions are periodic in Z and φ while special care = and H is the disk scale height. At R R0 approximate thermal has been paid to the radial boundaries. Here vR, vz, Bφ, Bz are set σ equilibrium, Eq. (15), can be used to express as a function of to zero, BR is computed to enforce magnetic flux conservation, vφ disk parameters: is set to the Keplerian value, and finally temperature and density are fixed to their initial values. 3 αR2Ω3 H 2 σ ≈ 0 0 0 · (18) As is common in simulations of this kind, we create two 2 γ 4 R buffer zones adjacent to the inner and outer boundaries in which T0 0 the velocities are damped toward their boundary values in or- − Now setting α ∼ 10 2 (and moving to numerical units) ensures der to avoid sharp discontinuities. The buffer zones extend from that σ only depends on the ratio H0/R0 = c0/(R0Ω0). We inves- R = 1toR = 1.5 for the inner buffer and from R = 7.5toR = 8 tigate two cases: a “hot” disk with H0/R0 = 0.1 which yields for the outer one. For the same reason, a large resistivity is used −4 σ = 1.1 × 10 , and a “cold” disk with H0/R0 = 0.05 and thus in those buffer zones in order to prevent the magnetic field from − σ = 2.7 × 10 5. The two parameter choices for σ will be re- accumulating next to the boundary. As a result of this entire pro- ferred to as the σhot and σcold cases in the following. Using the cedure, turbulent activity decreases as one approaches the buffer relation between pressure and sound speed respectively yields zones. This would occasionally mean the complete absence of −3 −3 P0 = 7.1 × 10 and P0 = 1.8 × 10 for the two models. The turbulence in the region close to the inner edge because of inade- surface density profile in the simulations is given by a power quate resolution. To avoid that problem, the cooling parameter σ law: gradually increases with radius in the region R0 < R < 2R0.In p+q/2+3/2 Appendix A, we give for completeness the functional form of σ Σ=ρ R · 0H0 (19) 2 ∼ −2.5 PP ∼ −1.5 R0 With our choice of parameters, we have Q− R and Q− R . PP This means that the agreement between Q− and Q− improves with ra- In resistive simulations, the parameters Tmin, TMRI and η0 (when dius. At small radial distances, dust sublimates and our model breaks applicable) need to be specified for the runs to be completely down (see Sect. 6).

A22, page 4 of 15 J. Faure et al.: Dead zone inner edge dynamics we used. These parts of the domain that corresponds to the inner and outer buﬀers are hatched on all plots of this paper.

3. MHD simulations of fully turbulent PP disks We start by describing the results we obtained in the “ideal” MHD limit: η = 0 throughout the disk. As a consequence, there is no dead zone and the disk becomes fully turbulent as a result of the MRI. The purpose of this section is (a) to describe the thermal structure of the quasi-steady state that is obtained; (b) to check the predictions of simple alpha models; and (c) to examine the small-scale and short-time thermodynamic fluctuations of the gas, especially with respect to their role in turbulent heat diffusion. Once these issues are understood we can turn with confidence to the more complex models that exhibit dead-zones.

3.1. Long-time temperature profiles We discuss here the results of the two simulations that correspond to σhot and σcold. We evolve the simulations not only for a long enough time for the turbulence transport properties to reach a quasi-steady state but also for the thermodynamic properties to have also relaxed. Thus the simulations are evolved for a time much longer than the thermal time of the gas, τheat ≈ 25 local orbits. Here, we average over nearly 1000 inner orbits (about 2τheat at the outermost radius R = 7). In Fig. 1 we present the computed radial temperature profiles and the radial profiles of α for the two simulations. Note that α Fig. 1. Temperature profiles T averaged over 900 inner orbits (red is clearly not constant in space. In both models, α is of order curves). Black plain lines show their corresponding theoretical profiles. Black dashed lines show their corresponding initial profiles. Top a few times 10−2 and decreases outward. We caution here that panel: σcold case. Bottom panel: σhot case. The subframes inserted this number and radial profiles are to be taken with care. The in the upper right of each panel show the alpha profiles obtained in α value is well known to be affected by numerical convergence both cases (red lines). The dashed lines show the analytic profiles (Beckwith et al. 2011; Sorathia et al. 2012) as well as physical used to compute the theoretical temperature profile (see text), given by −2 −3 −2 −3 convergence issues (Fromang et al. 2007; Lesur & Longaretti α = 6.3 × 10 −7.1 × 10 R/R0 and α = 4.6 × 10 −5.7 × 10 R/R0 2007; Simon & Hawley 2009). This may also influence the tem- respectively. perature because it depends on the turbulent activity. The disk temperature T rapidly departs from the initialized to achieve thermal equilibrium. During the steady state phase of power law given by Eq. (11). In both plots the averaged simu- the simulations, the mass loss at the radial boundaries is very − . ρ ∼ × −4ρ Ω lated temperature decreases faster than R 0 5. This is partly a re- small: the restoring rate is ˙ 5 10 0 0 on average on the sult of α decreasing with radius, and a consequent reduced heat- domain. ing with radius. The disk aspect ratios are close to the targeted values: H/R ∼ 0.15 and 0.08 in model σhot and σcold respec- 3.2. Turbulent fluctuations of temperature tively, slightly increasing with radius in both cases. We now focus on the local and short time evolution of T,by One of our goals is to check the implementation of the investigating the fluctuations of density and temperature once thermodynamics, and to ensure that we have reached thermal the mean profile has reached a quasi steady state. We define the equilibrium. To accomplish this we compare the simulation tem- temperature fluctuations by δT = T(R,φ,Z, t)−T(R, t), where an perature profiles with theoretical temperature profiles computed overline indicates an azimuthal and vertical average. The magni- according to the results of Balbus & Papaloizou (1999). The the- tude of the fluctuations are plotted in Fig. 2.Intheσcold case, the oretical temperature profile can be deduced from thermal equi- fluctuations range between 4 and 8% while they vary between 3 librium, Eq. (15), in which we now include the full expression and 5% in the σhot case. The smaller temperature fluctuations in for the cooling function (i.e. including Tmin): the latter case possibly reflect the weaker turbulent transport in / the latter case, while also its greater heat capacity. The tendency 3 αΩP 1 4 T = T 4 + · (22) of the relative temperature fluctuations to decrease with radius is min 2 σρ due to α decreasing outward. These temperature fluctuations can be due to different kinds Using the simulated α profiles as inputs, we could then calculate of events that act, simultaneously or not, to suddenly heat or radial profiles for T. In fact, we used linear fits of α (shown as a cool the gas. Such events can, for example, be associated with dashed line) in Eq. (22), for simplicity. The theoretical curves are adiabatic compression or magnetic reconnection. To disentangle compared with the simulation results in Fig. 1. The overall good these different possibilities, we note that turbulent compressions, agreement validates our implementation of the source term in the being primarily adiabatic, satisfy the following relationship energy equation and also demonstrates that we accurately cap- between δT and the density fluctuations δρ: ture the turbulent heating. It is also a numerical confirmation that turbulent energy is locally dissipated into heat in MRI-driven tur- δT δρ ffi = (γ − 1) · (23) bulence and that our simulations have been run su ciently long T ρ A22, page 5 of 15 A&A 564, A22 (2014)

Fig. 2. Temperature standard deviation profiles averaged over 900 inner orbits. The plain line show the result from the σ case and the dashed Fig. 4. Thermal diffusivity’s radial profile averaged over 900 inner or- hot ff curve show the result from the σcold run. bits. Dashed lines show the mean thermal di usivity for two homo- geneous αT (αT = 0.02 and αT = 0.004). The grey area delimit one standard deviation around the mean thermal diffusivity.

By analogy with molecular thermal diffusivity, we quantify the turbulent efficiency to diffuse heat using a parameter κT : 2 κT ρc ∂T F = − 0 · (25) turb γ(γ − 1) ∂R To make connection with standard α disk models, we introduce the parameter αT which is a dimensionless measure of κT : κT αT = , (26) csH

Fig. 3. Blue dots map the scaling relation between temperature and den- as well as the turbulent Prandtl number PR = α/αT that com- sity fluctuations δT and δρ in the σhot simulation. 900 events are ran- pares turbulent thermal and angular momentum transports. We ff domly selected in the whole domain (except bu er zones) among 900 measure the turbulent thermal diffusivity in the σhot model and inner orbits. The red line shows the linear function (slope = γ − 1)) plot in Fig. 4 its radial profile and statistical deviation. For com- corresponding to the adiabatic scaling. parison, we also plot two radial profiles of κT that would result from constant αT , chosen such that they bracket the statistical de- In Fig. 3 we plot the distribution of a series of fluctuation events viations of the simulations results. They respectively correspond to PR ∼ 0.8andPR ∼ 3.5 (assuming a constant α = 0.02). The randomly selected during the σhot model in the (δρ/ρ, δT/T) plane. The dot opacity is a function of the event’s radius: darker large deviation around the mean value prevents any definitive points stand for outer parts of the disk. The events are scattered and accurate measurement of PR, but our data suggest that PR is around unity. This is relatively close to PR = 0.3thatPierens around the adiabatic slope (represented with a red line), but dis- ff play a large dispersion that decreases as radius increases. The et al. (2012) used to investigate, with a di usive model, how the ff turbulent diffusivity impacts on planet migration. di erence between the disk cooling time and its orbital period ff is so large that the correlation should be much better if all the Turbulent di usion must be compared to the radiative dif- heating/cooling events were due to adiabatic compression. This fusion of temperature. In the optically thick approximation, the ff ∝ 3 implies that temperature fluctuations are mainly the result of iso- radiative di usivity is T . One can establish, using Eq. (15), that the dimensionless measure of radiative diffusivity αrad,de- lated heating events such as magnetic reconnection. This sugges- α α tion is supported by the larger dispersion at short distance from fined similarly to T (see Eqs. (25)and(26)), is equal to a few . the star, where turbulent activity is largest. However, a detailed The turbulent transport of energy is then comparable to radiative study of the heating mechanisms induced by MRI turbulence is transport in PP disks (see also Appendix A in Latter & Balbus beyond the scope of this paper. Indeed, the low resolution and the 2012). Nevertheless, radiative transport is neglected in our sim- uncontrolled dissipation at small scales – important for magnetic ulations. As indicated by the discussion above, this shortcoming reconnection – both render any detailed analysis of the problem should be rectified in future. Note that radiative MHD simula- difficult. Future local simulations at high resolution and with ex- tions are challenging and Flock et al. (2013) performed the first plicit dissipation will be performed to investigate that question global radiative MHD simulations of protoplanetary disks only further. very recently. Previous studies had been confined to the shearing box approximation (Hirose et al. 2006; Flaig et al. 2010; Hirose As shown in Balbus (2004), MHD turbulence may also in- ffi duce thermal energy transport through correlations between the & Turner 2011) because of the inherent numerical di culties of temperature fluctuations δT and velocity fluctuations δu.Forex- radiative simulations. ample, the radial flux of thermal energy is given by Eq. (5) of Balbus (2004): 4. MHD simulations of PP disks with a dead zone ρc2 We now turn to the non-ideal MHD models in which the disk is F = 0 δTδv . (24) turb γ − 1 R composed of an inner turbulent region and a dead zone at larger A22, page 6 of 15 J. Faure et al.: Dead zone inner edge dynamics

Fig. 5. z = 0 planar maps of radial magnetic fields fluctuation (on the left) and density fluctuations (right plot) from the σhot simulation at t = 900 + τcool(R = 7R0). radii. To make contact with previous work (Dzyurkevich et al. 2010; Lyra & Mac Low 2012), we first consider the case of a resistivity that is only a function of position. Such a simplification is helpful to understand the complex thermodynamics of the dead zone before moving to the more realistic case in which the resistivity is self-consistently calculated as a function of the temperature using Eq. (8).

4.1. The case of a static interface

At t = 900, model σhot has reached thermal equilibrium. We then set η = 0forR < 3.5, and η = 10−3 for R ≥ 3.5, and restart the simulation. The higher value of η corresponds to a magnetic Reynolds number Rm ∼ csH/η ∼ 10. It is sufficient to stabilize the MRI, and as expected, the flow becomes laminar outward of R = 3.5 in a few orbits. The structure of the flow after 460 inner orbits (which roughly amounts to one cooling time at R = 7) is shown in Fig. 5. The left panel shows a snapshot Fig. 6. Pressure profile averaged over 200 orbits after t = 900+τ (R = of the radial magnetic field. The turbulent active region displays cool 7) in σHot case. The black solid line show the pressure profile 200 inner large fluctuations which can easily be identified. The interface orbits after freezing the temperature in the domain. The black dashed between the active and the dead zone, defined as the location line plots the radial temperature profile in the ideal case (i.e. without a where the Maxwell stress (or, equivalently, the magnetic field dead zone). fluctuations) drops to zero, stands at R = 3.5 until the end of this static non-ideal simulation. As soon as the turbulence vanishes, the sharp gradient of az- Ohmic heating, since the magnetic energy is extremely small imuthal stress at the interface drives a strong radial outflow. As outward of R = 3.5(atR = 4, the magnetic energy has dropped identified by Dzyurkevich et al. (2010)andLyra & Mac Low by a factor of 102 compared to R = 3.5, while at R = 5 it has been (2012), we observe the formation of a pressure and a density reduced by about 103). In contrast, there are significant hydrody- maximum near the active/dead interface. In Fig. 6,wepresent namic fluctuations in the dead zone, as shown on the right panel the mean pressure perturbation we obtained in that simulation. of Fig. 5. For example, density fluctuations typically amount to The growth time scale of that structure is so short that the source δρ/ρ of the order of a few percent at R = 5. The spiral shape of term in the continuity equation (see Sect. 2.1) has no qualita- the perturbation indicate that these perturbations could be den- tive impact because it acts on the longer time scale τρ.This sity waves propagating in the dead zone, likely excited at the feature persists even after we freeze the temperature and use dead/active interface. Such a mechanism would be similar to the a locally isothermal equation of state as shown by the black excitation of sound waves by the active surface layers observed solid line. We conclude that pressure maxima are robust features in vertically stratified shearing boxes simulations (Fleming & that form at the dead zone inner edge independently of the disk Stone 2003; Oishi & Mac Low 2009) and also the waves seen thermodynamics. in global simulations such as ours (Lyra & Mac Low 2012)but We next turn to the temperature’s radial profile. Because we performed using a locally isothermal equation of state. However, expect the dead zone to be laminar, we might assume the tem- as opposed to Lyra & Mac Low (2012), the Reynolds stress as- perature to drop to a value near Tmin = 0.05 (see Sect. 2.3). sociated with this waves is smaller in the dead zone than in the However, the time averaged temperature radial profile plotted in active zone, and amounts to α ∼ 10−4. Such a difference is likely Fig. 7 shows this is far from being the case. The temperature due to the smaller azimuthal extent we use here, as it prevents in the bulk of the dead zone levels off at T ∼ 0.3, significantly the appearance of a vortex (see Sect. 5.1 and Lyra & Mac Low above Tmin. This increased temperature cannot be attributed to 2012).

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Fig. 8. Turbulent thermal flux profile averaged over 200 inner orbits after t = 900 + τcool(R = 7) from the σhot run. The central dotted region locates the percolation region behind active/dead zone interface. Fig. 7. The radial profile of temperature (solid red line) averaged over 200 orbits after t = 900 + τcool(R = 7R0). The red dashed line describes the temperature profile from the ideal case and the wave-equilibrium certainly supports the assumption that waves are generated at profile is shown by the plain black line. The black dashed lines show the interface. the threshold value TMRI used in Sect. 4.3 and (5/4)TMRI. The grey area shows the deviation of temperature at 3 sigmas. The vertical line shows In order to make the comparison more precise, we plot in the active/dead zone interface. Fig. 7 the radial profile of the temperature, computed using Eq. (28)3. The agreement with the temperature we measure in the simulation is more than acceptable in the bulk of the dead 4.2. Wave dissipation in the dead-zone zone. There are however significant deviations at R ≥ 7, i.e. next to the outer boundary. Associated with these deviations, we also It is tempting to use Eq. (22) to estimate the temperature that measured large, but non wave-like values of the quantity δρ/ρ at would result if the wave fluctuations were locally dissipated into ∼ that location (see Fig. 5). heat (as is the case in the active zone). However, we obtain T ff 0.1 which significantly underestimates the actual temperature. In order to confirm that the outer-bu er zone is responsible This is probably because the density waves we observe in the for these artefacts, we ran an additional simulation with identical dead zone do not dissipate locally. We now explore an alternative parameters but with a much wider radial extent. The outer-radial ff boundary is located at R = 22, and thus the outer-buffer zone model that describes their e ect on the thermodynamic structure = . = of the dead zone. extends from R 21 5toR 22. The additional simulation We assume that waves propagate adiabatically before dissi- shows that indeed the suspicious temperature bump moves to pating in the form of weak shocks in the dead zone; this assump- the outer boundary again. The temperature in the bulk of the ∼ dead zone, on the other hand, remains unchanged and is in good tion is consistent with measured Mach numbers ( 0.1) in the σ bulk of our simulated dead zone. A simple model for the weak agreement with the model hot. Further details of this calculation + are given in Appendix C. shocks, neglecting dispersion, yields a wave heating rate Qw: In Appendix C we show that the density wave theory can 3 + γ(γ + 1) δρ also account for the density fluctuations as a function of radius. Q = P f, (27) As waves propagate, their amplitudes decrease, not least because w 12 ρ their energy content is converted into heat. Overall, the good where f stands for the wave frequency (Ulmschneider 1970; match we have obtained between the wave properties’ radial pro- Charignon & Chièze 2013). The derivation of that expression files (both for the temperature and density fluctuations) and the is presented in Appendix B. Balancing that heating by the lo- analytical prediction strongly favour our interpretation that den- cal cooling rate ρσT 4 yields an estimate of the temperature’s sity waves are emitted at the active/dead zone interface and rule radial profile. out any numerical artefacts that might be associated with the domain outer boundary. γ + 1 H 2 δρ 3 The effect of the waves can be viewed as a flux of thermal σT 4 = (RΩ)2 f, (28) 12 R ρ energy carried outward. Such a thermal flux can be quantified using Eq. (24) and its radial profile in model σhot is shown in where the relation between the pressure and the disk scale height Fig. 8. It is positive and decreasing in the dead zone which con- has been used. This expression provides an estimate of the dead- firms that thermal energy is transported from the active zone and zone temperature. If one assumes that the waves are excited deposited at larger radii. While it is rapidly decreasing with R, around the dead-zone interface, the frequency should be of order the important point of Fig. 8 is that it remains important in a per- the inverse of the correlation time τc of the turbulent fluctuations colation region outward of the dead zone inner edge (located at at that location. Such a short period for the waves, much shorter R = 3.5). Most of the wave energy that escapes the active region than the disk cooling time, is consistent with the hypothesis that is transmitted to the dead zone over a percolation length,thesize these waves behave adiabatically. Local simulations of Fromang of the percolation region. We define the percolation region as the & Papaloizou (2006) suggest τc/Torb ∼ 0.15. Given the units of region inside the dead zone where the thermal transport amounts the simulations, this means that we should use f ∼ 1. Taking δρ/ρ ∼ 0.1andH/R ∼ 0.1, this gives T ∼ 0.25, at R = 5, which 3 We used the azimuthally and temporally averaged simulation data for agrees relatively well with the measured value. The agreement H/R and δρ/ρ when computing the temperature using Eq. (28).

A22, page 8 of 15 J. Faure et al.: Dead zone inner edge dynamics

Fig. 9. Maps of radial magnetic ﬁeld at diﬀerent time t = 1600 + t . t is given in inner orbital time.

Fig. 10. Space-time diagrams showing the turbulent activity evolution in the MHD simulation (left panel) and in the mean field model (right panel). In the case of the MHD simulation, α is averaged over Z and φ. In the case of the mean field model, the different white lines mark the front position for the additional models described in Sect. 4.3. The solid line represents the case of remnant heating in the dead zone, the dashed line no remnant heating, and the dotted line the case of constant thermal diffusivity over the whole domain. Note the different vertical scales of the two panels (see discussion in the text for its origin). to more that half its value in the active zone. As shown by Fig. 8, it retains its coherence. However, as shown by the third panel of the percolation length l ∼ 0.5 in model σhot, which translates to Fig. 9, the turbulent front does not stop at R ∼ 4.5 but moves out- about one scale height at that location. ward all the way to R ∼ 5.5–6. This is confirmed by the left panel Though the heat flux is greatest within about H of the dead of Fig. 10, in which a space-time diagram of α indicates that the zone interface, the action of the density waves throughout our front reaches its stagnation radius in a few hundred orbital peri- simulated dead zone keeps temperatures significantly higher ods4. The following questions arise: can we predict the stalling than what would be the case in radiative equilibrium. This would radius? And can we understand this typical time scale? What can indicate that much of the dead zone in real disks would be hotter it tell us about the physical mechanism of front propagation? than predicted by global structure models that omit this heating source. In particular, this should have important implications Stalling radius: we use the mean field model proposed by Latter for the location of the ice line, amongst other important disk &Balbus(2012) to interpret our results. The front-stalling radius features. can be calculated from the requirement that the integrated cooling and heating across the interface balance each other: 4.3. The case of a dynamic interface TA (Q+ −L)dT = 0 (29) We now move to the main motivation of this paper. At t = 1600 TDZ we restart model σhot but close the feedback loop between turbulence and temperature. The resistivity η is now given by Eq. (8) in which TDZ and TA stand for the temperature in the dead zone with η = 10−3. We set the temperature threshold to the value and in the active zone respectively. While the cooling part of the 0 + TMRI = 0.4, i.e. slightly above the typical temperature in the integral is well defined, the heating part Q is more uncertain. dead zone (which is about 0.25 as discussed above). This set-up It is probably a mixture of turbulent heating (with an effective ensures that at least half of the radial domain is “bistable”, i.e. α parameter that varies across the front) and wave heating such gas at the same radius can support either one of two quasi-steady as described in Sect. 4.1. For simplicity, we adopt the most naive stable states, a laminar cold state, and a turbulent hot state. 4 To check that Eq. (8) is an acceptable simplification of the actual As seen in Fig. 7, the temperature exceeds TMRI in the region exponential dependence of η with temperature, we ran an additional 3.5 ≤ R ≤ 4.5 at restart. Thus, we would naively expect the in- simulation, with the same parameters as for the σhot but using Eq. (7) terface to move to R ∼ 4.5. As shown by snapshots of the radial for η. We find no difference with our fiducial model: the front propa- magnetic field fluctuations in the (R,φ) plane at three different gates equally fast and stops at R 5.3, i.e. almost the same radius as times in Fig. 9, indeed the front moves outwards. In doing so, for the σhot case.

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Fig. 11. α profiles averaged over 100 orbits after the fronts have reached their final position (α values are listed on the left axis). Left panel: σhot simulation which TMRI = 0.46. Middle and right panels: σcold simulation which TMRI = 0.1andTMRI = 0.13 respectively. The black plain lines remind the temperature profiles from ideal cases. Black dashed lines show the threshold value TMRI and 5/4TMRI. The temperature values are listed on the right axis. approach possible and assume that Q+ is a constant within the We solve the partial differential equation for the temperature (we front (equal to 1.5ΩTRφ) as long as the temperature T is larger have dropped the overbar here for clarity): than T , and vanishes otherwise. In that situation, we obtain, MRI ∂ γ γ − σ κ ∂ ∂ T ( 1) 4 4 T T similarly to Latter & Balbus (2012) that the stalling radius Rc is = 1.5ΩT φ − (T − T ) + R · (31) ∂t R 2 min R ∂R ∂R determined by the implicit relation: c0 5 TA(Rc) = TMRI. (30) To solve that equation, we have used the heating term measured 4 during the ideal MHD simulation σhot (see Sect. 3). The ther- In Fig. 7, we draw two horizontal lines that correspond to T = mal transport of energy is modeled by a simple diffusive law TMRI and T = (5/4)TMRI. The condition given by Eq. (30)is that is supposed to account for turbulent transport. As discussed satisfied for R ∼ 5.25 which is very close to the critical radius in Sect. 4.1, the dead zone is heated by waves excited at the actually reached in simulations (see Figs. 9 and the left panel of dead/active interface. It is thus highly uncertain (and one of Fig. 10). the purpose of this comparison) whether a diffusive model ad- In order to investigate the robustness of this result, we have equately describes heat transport of this fashion. The value for carried a further series of numerical experiments. First, we have κT is chosen so heat diffusion gives the same mean flux of ther- repeated model σhot using a value of TMRI = 0.46 (instead of mal energy as measured in our simulation (shown in Fig. 8). We TMRI = 0.4). According to Fig. 7, we expect the interface now have simplified the radial profile in that plot and have assumed to propagate over a smaller distance. As shown by the left hand that the thermal flux vanishes outward of the percolation region side panel of Fig. 11, this is indeed the case: the front stalls at but is uniform in the active zone and in the percolation region: ∼ = / R 5 which is precisely the position where T (5 4)TMRI. κ < + We also computed two analogues of those models with σ . κ = 0 if R Rf(t) Lp cold . (32) This was done as follows: after the σcold ideal MHD simulations 0otherwise has reached a quasi thermal equilibrium (t = 900), we intro- duced a static dead zone from R = 3.5 to the outer boundary of The front location R f (t) is evaluated at each time step. It is the T < T the domain and we restarted this simulation. We waited for ther- smallest radius where MRI. As in the MHD simulation, we T = . T = . L = mal equilibrium to close the feedback loop at t = 1600. We used used MRI 0 4and min 0 05. When p 0, we have found the dead zone inner edge is static regardless of the value of κ0.As TMRI = 0.1. A front propagates until the critical radius R ∼ 6.5. It is as fast as the front of the σ case. We show in the middle expected, the front displacement requires the active and the dead hot zone to be thermally connected. We have thus used a percolation panel of Fig. 11 that, in accordance with the previous result, the = . κ = . × −4 front stops close to the critical radius predicted by Eq. (30). We length Lp 0 5 and a value of 0 2 5 10 in the active part of the disk which matches the thermal flux at the outer edge of the repeated the same experiment using TMRI = 0.13 and show on the right panel of Fig. 11 that the front stalling radius is again percolation region (see Fig. 8). The simulation is initialized with accounted for by the same equation. the same physical configuration as the MHD simulation: a dead R = . R = To summarize, the front stops close to the predicted value zone extends from 3 5to 8 and is in thermal equilibrium. As shown on the right panel of Fig. 10, we find that a front in every simulations we performed. This shows that the equi- . − librium position of a dead zone inner edge can be predicted as propagates outward. The critical radius Rc 5 5 6isinagree- a function of the disk’s radiative properties and thermal struc- ment with the results of the simulation and with the argument based on energy conservation detailed above. However, there is ture. It is robust, and in particular does not depend on the details ff of the turbulent saturation in our simulations (such as the radial clearly a di erence in the typical time scale of the propagation: α profile). the front observed in the MHD simulation propagates five time faster than the front obtained in the mean field simulation. In order to check the sensitivity of this results to some of the Front speed and propagation: the previous result is based on uncertain aspects of the mean field model, we have run three ad- dynamical systems arguments. A front is static at a given radial ditional mean field simulations similar to that described above. location if the attraction of the active turbulent state balances In these additional models, we change one parameter while the attraction of the quiescent state. However, the argument does keeping all others fixed. In the first model, we have changed the not help us identify the front propagation speed. In order to un- minimum value of the temperature Tmin = 0.3 (this is a crude derstand its dynamics, we ran a set of mean field simulations way to model the effect of the dead zone heating at long dis- akin to the “slaved model” in Latter & Balbus (2012) but us- tances from the interface). In the second model, we have used −4 ing parameters chosen to be as close as possible to model σhot. κ0 = 5 × 10 . In the last model, we have used Lp =+∞,

A22, page 10 of 15 J. Faure et al.: Dead zone inner edge dynamics

Fig. 12. Vortensity ((V − Vinit)/Vinit) map in the disk (R,φ) plane. The left panel shows the midplane vortensity in the σhot simulation with the small azimuthal extend. The right panel shows the midplane vortensity in the σhot simulation which has φ ∈ [0,π/2]. thereby extending turbulent heat transport to the entire radial ex- pressure bump forms at the interface which then triggers the tent of the simulation. The results are shown on the right panel formation of a vortex. It is natural to wonder how these re- of Fig. 10: in all cases, the front propagates outward slower than sults are modified when better account is made of the gas observed in the MHD simulation. thermodynamics. A slow front velocity is thus a generic feature of the diffusive In our simulations of a static dead zone (Sect. 4.1), we also approximation to heat transport. But conversely, it reveals that find that pressure maxima form. Even with a T dependent η, the fast fronts mediated by real MRI turbulence are controlled the bumps survive as the interface travels to its stalling radius. by non-diffusive heat transport. This in turn strongly suggests Moreover, the amplitude of the pressure bump is not modified that the front moves forward via the action of fast non-local den- compared to those observed in simulations of static dead zones. sity wave heating, and not via the slow local turbulent motions On account of the reduced azimuthal extend of our domains, no of the MRI near the interface, as originally proposed by Latter & large-scale vortices formed. (The reduced azimuth was chosen Balbus (2012). The transport via density waves occurs at a veloc- so as to minimize the computational cost of our simulations.) In ity of order cs.Thus,ittakesatimeΔtwaves ∼ H/cs τorb/2π for order to observe the development of the Rossby wave instabil- thermal energy to be transported through the percolation length ity we performed one run identical to the σhot simulation except Lp ∼ H. This is shorter than the typical diffusion time over the we extended the azimuthal domain: φ = [0,π/2]. The η was a 2 same distance Δtdiff = H /κ > 100/2πτorb at R = 3.5, for the given function of position as in Sect. 4.1, thus the dead/active values of κ used in Eq. (31). zone interface was fixed. The right panel of Fig. 12 shows a late snapshot of this run, in which a vortex has appeared similar to those observed by Lyra & Mac Low (2012). It survives for 5. Instability and structure formation many dynamical time scales. Consistent with the results of Lyra α . Other issues that can be explored through our simulations are in- &MacLow(2012), we measure 0 01 in the dead zone. This σ stability and structure formation at the dead/active zone interface is two orders of magnitude larger than in the hot model. The and throughout the dead-zone. The extremum in pressure at the dead zone is consequently much hotter in this simulation, which interface is likely to give rise to a vortex (or Rossby wave) in- means the vortex plays a crucial role in both accretion and the stability (Lovelace et al. 1999; Varnière & Tagger 2006; Meheut thermal physics of the dead zone. Simulations are currently un- et al. 2012). On the other hand, the interface will control both the derway to investigate the robustness of this results and the vor- η = η midplane temperature and density structure throughout the dead tex survival when we close the feedback loop (setting (T)). zone; it hence determines the magnitude of the squared radial This will be the focus of a future publication. Brunt-Väisälä frequency NR. The size and sign of this important disk property is key to the emergence of the subcritical baro- 5.2. Subcritical baroclinic and double-diffusive instabilities clinic instability and resistive double-diffusive instability in the dead zone (Lesur & Papaloizou 2010; Latter et al. 2010; Klahr Another potentially interesting feature of the interface is the & Bodenheimer 2003; Petersen et al. 2007b,a). In this final sec- strong entropy gradient that might develop near the interface. tion we briefly discuss these instabilities, leaving their detailed It could also impact on the stability of the flow on shorter scales, numerical analysis for a future paper. giving rise potentially to the baroclinic instability (Lesur & Papaloizou 2010; Raettig et al. 2013) or to the double-diffusive instability (Latter et al. 2010). Both instabilities are sensitive to 5.1. Rossby wave instability the sign and magnitude of the entropy gradient, which is best The Rossby wave instability has been studied recently by Lyra quantified by the Brunt-Väisälä frequency NR: &MacLow(2012) with MHD simulations that use a locally isothermal equation of state and a static dead zone. Vortex for- 1 ∂P ∂ P N 2 = − ln · (33) mation mediated by the Rossby wave instability was reported: a R γρ ∂R ∂R ργ A22, page 11 of 15 A&A 564, A22 (2014)

2 In general, we find that NR takes positive values of or- is the key ingredient for the development of both the subcritical der 10−20% of the angular frequency squared. Negative val- baroclinic instability and the resistive double-diffusive instabil- ues sometimes appear localized next to the interface. Positive ity. We find that NR can take both positive and negative values values rule out both the baroclinic instability and the double- at different radii; but we caution that these preliminary results diffusive instability, and indeed, we see neither in the simula- require more testing with dedicated simulations. tions. However, we caution against any premature conclusions Several improvements are possible and are the basis of fu- about the prevalence of these instabilities in real disk. First, the ture work. As discussed in Sect. 5, one obvious extension is to source term in the continuity equation might alter the density investigate the fate and properties of emergent vortices at the radial profile and, consequently, the Brunt-Väisälä frequency in dead-zone inner edge. This can be undertaken with computa- the dead zone. Second, it is well known that these instabilities tional domains of a wider azimuthal extent. Such simulations are sensitive to microscopic heat diffusion. We do not include may investigate the role of large-scale vortices, and the waves such a process explicitly in our simulations. Instead, there is nu- they generate, on the thermodynamic structure of the dead-zone. merical diffusion of heat by the grid, the nature of which may They can also observe any feedback of the thermodynamics on be unphysical. Both reasons preclude definite conclusions at this vortex production and evolution. We also plan to investigate stage. Dedicated and controlled simulations are needed to assess other magnetic field configurations. For example, vertical mag- the existence and nonlinear development of these instabilities in netic fields might disturb the picture presented here because of the dead zone. the vigorous channel modes that might develop in the marginal gas at the dead-zone edge (Latter et al. 2010). Such an environment may militate against the development of pressure bumps 6. Conclusion and/or vortices. In this paper, we have performed non-ideal MHD simulations re- Our results, employing the cylindrical approximation, repre- laxing the locally isothermal equation of state commonly used. sents a thin region around the PP disk midplane. These results We have shown the active zone strongly influences the thermo- must be extended so that the vertical structure of the disk is in- dynamic structure of the dead zone via density waves gener- corporated. An urgent question to be addressed is the location of ated at the interface. These waves transport thermal energy from density wave dissipation in such global models. Waves can re- the interface deep into the dead zone, providing the dominant fract in thermally stratified disks and deposit their energy at the heating source in its inner 20H. As a consequence, the temper- midplane or in the upper layers depending on the type of wave ature never reaches the very low level set by irradiation. In the and the stratification (Bate et al. 2002, and references therein). outer regions of the dead-zone, however, temperatures will be set In particular, Bate et al. demonstrate that large-scale axisymet- by the starlight reprocessed by the disk’s upper layers (Chiang ric (and low m) density waves (fe modes) propagate upwards, as & Goldreich 1997; D’Alessio et al. 1998). It is because the well as radially, until they reach the upper layers of the disk, at wave generation and dissipation is located at the midplane that which point they transform into surface gravity waves and prop- waves should so strongly influence the thermodynamic struc- agate along the disk surface. But this is only shown for vertically ture of the dead zone. Note also that density waves generated polytropic disks in which the temperature decreases with z and by the dead/active zone edge are stronger than those excited by for waves with relatively low initial amplitudes. In PP-disk dead the warm turbulent upper layers of the disk as seen in stratified zones we expect the opposite to be the case, and it is uncertain shearing box (Fleming & Stone 2003). how density waves behave in this different environment. Another result of this paper concerns the dynamical be- Finally, in this work we have increased the realism of one haviour of the dead-zone inner edge. We find the active/dead aspect of the physical problem, the dynamics of the turbulence, interface propagates over several H (i.e. a few tens of an au) in (via direct MHD simulations of the MRI), but have greatly a few hundreds orbits. All the simulated MRI fronts reached a simplified the physics of radiative cooling. As a result, the final position that matched the prediction made by a mean field simulations presented here are still highly idealized and several approach (Latter & Balbus 2012), which appeals to dynamical improvements should be the focus of future investigations. For systems arguments. As the gas here is bistable, it can fall into example, our approach completely neglects the fact that dust either a dead or active state; the front stalls at the location where sublimates when the temperature exceeds ∼1500 K. As a re- the nonlinear attraction of the active and dead states are in bal- sult, the opacity drops by up to four orders of magnitude with ance. In contrast, the mean-field model fails to correctly predict potentially dramatic consequences for the disk energy budget the velocity of the simulated fronts. We find that a diffusive de- (leading to a increase of the cooling rate Q− of the same or- scription of the radial energy flux yields front speeds that are der, as opposed to our assumption of a constant σ). The radial too slow. In fact, the simulations show that fronts move rapidly temperature profile of our simulations indicates that the sublima- via the efficient transport of energy by density waves across the tion radius should be located 3−5 disk scaleheights away from interface. Fronts do not propagate via the action of the slower the turbulent front. Given that the dominant dynamical process MRI-turbulent motions. we describe here is mediated by density waves characterized by In addition, we have used our simulations to probe the ther- a fast time scale (compared to the turbulent and radiative time mal properties of turbulent PP disks. We have constrained the scales), we do not expect the front dynamics to be completely turbulent Prandtl number of the flow to be of order unity. We altered. Nevertheless, the relative proximity between the subli- have also quantified the turbulent fluctuations of temperature: mation radius and the turbulent front is still likely to introduce they are typically of order a few percent of the local temperature. quantitative changes. Clearly, the thermodynamics of that re- However, their origin – adiabatic compression vs. reconnection – gion is more intricate than the simple idealized treatment we use is difficult to assess using global simulations. In-depth dedicated in this work. This further highlights the need, in future work, non-isothermal shearing-box simulations will help to distinguish for a more realistic treatment of radiative cooling (for example the dominant cause of the temperature fluctuations. Finally, we using the flux limited diffusion approximation with appropriate have made a first attempt to estimate the radial profile of the ra- opacities, in combination with vertical stratification. Such simu- dial Brunt-Väisälä frequency NR in the dead zone. This quantity lations will supersede our heuristic cooling law with its constant

A22, page 12 of 15 J. Faure et al.: Dead zone inner edge dynamics

σ parameter. They are an enormous challenge, but will be essen- Hawley, J. F. 2001, ApJ, 554, 534 tial to test the robustness of the basic results we present here. Hirose, S., & Turner, N. J. 2011, ApJ, 732, L30 Hirose, S., Krolik, J. H., & Stone, J. M. 2006, ApJ, 640, 901 Acknowledgements. J.F. and S.F. acknowledge funding from the European Igea, J., & Glassgold, A. E. 1999, ApJ, 518, 848 Research Council under the European Union’s Seventh Framework Programme Klahr, H. H., & Bodenheimer, P. 2003, ApJ, 582, 869 (FP7/2007−2013)/ERC Grant agreement No. 258729. H.L. acknowledge sup- Kretke, K. A., Lin, D. N. C., Garaud, P., & Turner, N. J. 2009, ApJ, 690, 407 port via STFC grant ST/G002584/1. The simulations presented in this paper were Landau, L. D., & Lifshitz, E. M. 1959, Fluid Mech., 385 granted access to the HPC resources of Cines under the allocation x2012042231 Latter, H. N., & Balbus, S. 2012, MNRAS, 424, 1977 and x2013042231 made by GENCI (Grand Equipement National de Calcul Latter, H. N., Bonart, J. F., & Balbus, S. A. 2010, MNRAS, 405, 1831 Intensif). Lesur, G., & Longaretti, P.-Y. 2007, MNRAS, 378, 1471 Lesur, G., & Papaloizou, J. C. B. 2010, A&A, 513, A60 Lovelace, R. V. E., Li, H., Colgate, S. A., & Nelson, A. F. 1999, ApJ, 513, 805 References Lyra, W., & Mac Low, M.-M. 2012, ApJ, 756, 62 Lyra, W., Johansen, A., Zsom, A., Klahr, H., & Piskunov, N. 2009, A&A, 497, Armitage, P. J. 1998, ApJ, 501, L189 869 Balbus, S. A. 2004, ApJ, 600, 865 Meheut, H., Meliani, Z., Varniere, P., & Benz, W. 2012, A&A, 545, A134 Balbus, S. A. 2011, Magnetohydrodynamics of Protostellar Disks, ed. P. J. V. Nelson, R. P., & Gressel, O. 2010, MNRAS, 409, 639 Garcia (Chicago: University of Chicago Press), 237 Oishi, J. S., & Mac Low, M.-M. 2009, ApJ, 704, 1239 Balbus, S. A., & Hawley, J. F. 1998, Rev. Mod. Phys., 70, 1 Perez-Becker, D., & Chiang, E. 2011, ApJ, 735, 8 Balbus, S. A., & Hawley, J. F. 2000, in From Dust to Terrestrial Planets (Kluwer Petersen, M. R., Julien, K., & Stewart, G. R. 2007a, ApJ, 658, 1236 Academic), 39 Petersen, M. R., Stewart, G. R., & Julien, K. 2007b, ApJ, 658, 1252 Balbus, S. A., & Papaloizou, J. C. B. 1999, ApJ, 521, 650 Pierens, A., Baruteau, C., & Hersant, F. 2012, MNRAS, 427, 1562 Barge, P., & Sommeria, J. 1995, A&A, 295, L1 Pneuman, G. W., & Mitchell, T. P. 1965, Icarus, 4, 494 Baruteau, C., Fromang, S., Nelson, R. P., & Masset, F. 2011, A&A, 533, A84 Raettig, N., Lyra, W., & Klahr, H. 2013, ApJ, 765, 115 Beckwith, K., Armitage, P. J., & Simon, J. B. 2011, MNRAS, 416, 361 Sano, T., Miyama, S. M., Umebayashi, T., & Nakano, T. 2000, ApJ, 543, 486 Blaes, O. M., & Balbus, S. A. 1994, ApJ, 421, 163 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 Chambers, J. E. 2009, ApJ, 705, 1206 Simon, J. B., & Hawley, J. F. 2009, ApJ, 707, 833 Charignon, C., & Chièze, J.-P. 2013, A&A, 550, A105 Sorathia, K. A., Reynolds, C. S., Stone, J. M., & Beckwith, K. 2012, ApJ, 749, Chiang, E. I., & Goldreich, P. 1997, ApJ, 490, 368 189 D’Alessio, P., Canto, J., Calvet, N., & Lizano, S. 1998, ApJ, 500, 411 Steinacker, A., & Papaloizou, J. C. B. 2002, ApJ, 571, 413 Dzyurkevich, N., Flock, M., Turner, N. J., Klahr, H., & Henning, T. 2010, A&A, Teyssier, R. 2002, A&A, 385, 337 515, A70 Ulmschneider, P. 1970, Sol. Phys., 12, 403 Flaig, M., Kley, W., & Kissmann, R. 2010, MNRAS, 409, 1297 Umebayashi, T., & Nakano, T. 1988, Prog. Theor. Phys. Suppl., 96, 151 Fleming, T., & Stone, J. M. 2003, ApJ, 585, 908 Varnière, P., & Tagger, M. 2006, A&A, 446, L13 Flock, M., Fromang, S., González, M., & Commerçon, B. 2013, A&A, 560, A43 Zhu, Z., Hartmann, L., & Gammie, C. 2009a, ApJ, 694, 1045 Fromang, S., & Papaloizou, J. 2006, A&A, 452, 751 Zhu, Z., Hartmann, L., Gammie, C., & McKinney, J. C. 2009b, ApJ, 701, Fromang, S., Hennebelle, P., & Teyssier, R. 2006, A&A, 457, 371 620 Fromang, S., Papaloizou, J., Lesur, G., & Heinemann, T. 2007, A&A, 476, 1123 Zhu, Z., Hartmann, L., Gammie, C. F., et al. 2010, ApJ, 713, 1134

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A22, page 13 of 15 A&A 564, A22 (2014)

Appendix A: Opacity law in the buffer zones Here, we give the functional form of σ we used to prevent the temperature to drop at the inner edge of the simulation. ⎧ ⎪ −4 ⎨⎪σ − R < σ = 0 1 2R if R 2R0 ⎩⎪ 0 (A.1) σ0 otherwise, where σ0 stands for σhot or σcold depending on the model. The value of σ is kept constant in the outer buffer. Fig. B.1. Velocity fluctuations profile: series of “teeth” modelling the wave shocks. Appendix B: Wave heating The large-scale density waves witnessed in our simulations develop weak-shock profiles, which are controlled by a com- petition between nonlinear steepening and wave dispersion. In addition to that estimate, the thermal energy flux diver- Keplerian shear may also play a role as it “winds up” the spi- gence can be used, through Eq. (B.4), as a way to estimate the ral and decreases the radial wavelength; though by the time this radial variation of the wave amplitude: ⎡ ⎤ is important most of the wave energy has already dissipated. ⎢ 2⎥ 1 ∂RFR 1 ∂ ⎢γP c R δρ ⎥ A crude model that omits the strong dispersion inherent in = ⎣⎢ 0 s ⎦⎥ (B.6) our large-scale density waves nevertheless can successfully ac- R ∂R R ∂R 12 ρ count for the energy dissipation in the simulations. In such a ⎡ γP c ⎢ δρ 2 1 ∂P δρ 2 1 ∂c model the density wave profiles are dominated by steepening = 0 s ⎣⎢ 0 + s and can thus be approximated by a sawtooth shape propagating 12 ρ P0 ∂R ρ cs ∂R at the sound speed velocity cs (see Fig. B.1). The evolution of δρ ∂ δρ δρ 2 + + 1 · the amplitude of such isentropic waves is given by Landau & 2 ρ ∂ ρ ρ Lifshitz (1959). In the wave frame of reference, the gas velocity R R at the shock crest evolves over time as the shock wave dissipates: This must equal the energy rate released as thermal heat given v0 by −D. Combining the last two expression thus provides an ex- v(t) = , (B.1) (γ+1) pression for the radial decay of the wave amplitude: 1 + λ v0t 2 0 v λ ∂ δρ δρ 1 1 ∂P 1 ∂c γ + 1 δρ where 0 is the excitation amplitude of the wave and 0 its wave- = − 0 − s − f . (B.7) length, assumed to be conserved over the wave propagation. The ∂R ρ 2ρ R P0 ∂R P0 ∂R cs ρ mean mechanical energy embodied in one wave period at time t isgivenbyanintegraloverradius: The first term is the geometrical term that described the wave di- lution as it propagates cylindrically. The second and third terms λ / 2 ρ 0 2 are specific to waves propagating in stratified media where mean = R − v 2 = 1 ρv 2 Et 1 (t) dR (t) (B.2) pressure and sound speed are not uniform. Finally, the last term λ0 −λ / λ/2 12 0 2 of the right hand side of this equation reflects the wave damping where λ is the wavelength at time t. We next compute the me- by shocks. In our simulations, all four terms are of comparable chanical energy radial flux through a unit surface: importance. ρv(t)2c γP c δρ 2 F = E c = s = 0 s (B.3) R t s 12 12 ρ Appendix C: Model with an extended radial extent where δρ is the difference between the shocked and pre-shocked In this section of the appendix we describe the results obtained density. To compute the last equality, we have used the fact that, in the radially extended σhot run. We initiated MHD turbulence under the weak shock approximation, the wave evolution is isen- in that model in the absence of any dissipative term until the tropic and thus δρ/ρ = v/cs (Landau & Lifshitz 1959). The wave region between R = 1andR = 3.5 has reached thermal equilib- −3 energy and its flux are related through the following conserva- rium (t = 900). At that point, we set η0 = 10 for R ≥ 3.5. As tion law: expected, a dead zone quickly appeared at those radii. Because of the prohibitic computational coast of that simulation (there ∂E 1 ∂RF t + R = , are 960 cells in the radial direction!), we were not able to run ∂ ∂ 0 (B.4) t R R that model until thermal equilibrium is established at all radii. while the dissipation rate of the wave is expressed using the me- Indeed, the cooling time becomes very long at large radius. chanical energy conversion into heat per unit time: Instead, we followed the time evolution of the temperature at nine locations in the dead zone. Due to the absence of turbulent ∂E 1 ∂v γ(γ + 1) δρ 3 heating, we found that the temperature slowly decreases with D = t = ρv = P f (B.5) ∂t 12 ∂t 12 0 ρ time. Assuming this decrease is due to a combination of cooling (resulting from the cooling function) and wave heating, it can be where f = cs/λ0 is the wave frequency. We used the density fluc- modelled using the following differential equation tuations in our simulations as the difference between the shocked ∂T γ(γ − 1)σ and pre-shocked density to estimate the local wave heating in = − hot T 4 + Q+ . (C.1) Eq. (27). ∂t 2 w c0 A22, page 14 of 15 J. Faure et al.: Dead zone inner edge dynamics

+ where the term Qw accounts for the (unknown) wave heating. We fitted the time evolution of the temperature during the duration of the simulation (∼1000 orbits) in order to obtain a numerical + estimate of Qw at those nine locations. An example of that fit is provided in the insert of Fig. C.1. Using these value, we can obtain an estimate for the equilibrium temperature in the dead zone, as shown in Fig. C.1. The comparison with the estimate of Eq. (28) provided by the black line is excellent everywhere in the dead zone. As a sanity check, we test here if the wave amplitude decreases as their energy content is converted into heat. We show in Fig. C.2 the mean density fluctuation profile obtained in the extended σhot simulation. It exhibits two maxima close to the dead zone inner edge located at R(g,1) and R(g,2) and the ampli- Fig. C.2. Profile of density standard deviation δρ averaged over tude decay with radius. The reason why we see two such max- 200 inner orbits after t = 900 + τcool(R = 7) in the extended σhot case. ima is not clear but might be due to waves originated at the The black plain line shows the density fluctuation amplitude deducted dead/active interface as well as waves excited as the location from the two waves model. The excitation locations used in the two of the pressure maximum. In any case, we found that modelling waves model are shown by blue dashed lines. the amplitude of fluctuations as the signature of a combination of two waves generated at R(g,i) gives acceptable results. We Appendix D: Simulation with a higher resolution use an explicit scheme to numerically integrate Eq. (B.7) from the wave generation locations R , . The two waves are excited Here we present a brief test of the impact of spatial resolution on (g i) σ with the amplitude measured at R = R , with the frequency our results. We have restarted the hot run from the thermal equi- (g i) = 1/ f = 0.15τ (R , ). We plot the solution thus obtained in librium of the ideal MHD case (t 600) and the static dead zone cool (g i) = Fig. C.2. The good agreement between the analytical solution case (t 900), with twice as many cells in each direction. The , , and the profile gives a final confirmation that waves control the resolution is (640 160 160). We show the averaged temperature dead zone thermodynamics. profile of both cases in Fig. D.1. Because of the large computational cost, each models are integrated for 200 orbits and time averages are only performed over the last 100 orbits. The temperature profiles of each case are very close to those obtained with the fiducial runs. We conclude that resolution as little impact on our results.

Fig. C.1. Temperature profile T averaged over 200 orbits after t = 900 + τcool(R = 7) obtained in the extended σhot case. The black line show wave-equilibrium temperature profile. Red dots show the extrap- Fig. D.1. Profiles of temperature T averaged over 100 inner orbits af- olated temperature for 7 radii. The inserted frame show in red the tem- ter t = 700 and t = 1000 in the highly resolved σhot case (plain lines). perature evolution T after t = 900 at R = 9.8R0 in the extended σhot The dashed lines remind the temperature profiles obtained with the low simulation. On this subplot, the black plain line shows the accurate fit resolution run. For both resolutions, the temperature in the ideal case is and the black dashed lines show the two extreme fits used to determine shown by a red line and the temperature in the static dead zone case is the measurement error. shown by a black line.

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